The typical size of Hecke eigenvalue sums is $o(\sqrt{x})$

Let $\mathcal{H}_k$ denote the set of normalized holomorphic Hecke cusp forms of weight $k$ for the full modular group $ \mathrm{SL}_2(\mathbb Z)$. For each $f\in\mathcal{H}_k$, let $λ_f(n)$ denote the corresponding eigenvalue of the normalized Hecke operator $T_n$. We prove nontrivial upper bounds